Thread by @fgrosshans

Seated for a talk by Mio Murao, form @UTokyo_News_en on Higher order quantum operations of blackbox unitaries and causal structure of the blackboxes. #LTQI

@UTokyo_News_en Mio Murao ([村尾 美緒) uses quantum computer to understand physics. What we can do do and what we cannot do are physics problems.
#LTQI

@UTokyo_News_en Mio Murao ([村尾 美緒) more specificely looks at maps of maps: The computer F takes as input a map f (e.g. given as a black box) and a quanutm state ρ and outputs F(f)(ρ).
Here we restrict ourseves to unitary f=U, and F(U)=V also unitary.
#LTQI

Mio Murao ([村尾 美緒): Some useful V_U, many impossible perfectly
• replication U⊗U
• Inversion U⁻¹=U⁺
• “time reversat” U*
• Transpose U^T
• Control U |0⟩⟨0|⊗I + exp(iθ_U) |1⟩⟨1|⊗U
• quantum switch
• neutralization V_U=I
#LTQI

Mio Murao ([村尾 美緒) defines her supermap as a quantum comb acting on the Choi matrices of the maps. It defines a set of constraints, including some induced by the predefinite Causal orders. It leads to may no-go theorems.
#LTQI

Mio Murao ([村尾 美緒): The no-go have different reasons: C-unitary because of linearity, CP for U^T, causal order for U* and quantum switch.
Gate teleportation allows to make transpose, with probability 1/d². What is the optimal setting ?
#LTQI

Mio Murao ([村尾 美緒)’s approach:
1. Adding “nonlinear power”, i.e. multiple (finite) use of U but a single use of ρ
Or
2. root powers of U : apply U^(1/d). Weird in terms of computer science, but makes sense in physics, when U=exp(iHt)
#LTQI

Mio Murao ([村尾 美緒): Algorithm U^(⊗d-1)→U* which is deterministic and exact, going through the (d-1) antisymmetric subspace. If you know group theory it is trivial, through the tensor product representation SU(d)^(⊗d-1)
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Mio Murao ([村尾 美緒) have a probailistic protocol for the transpose, based on port based teleportation with failure <d²/d. A sequential protocol is exponetially better (exp(-N/d³).
See arXiv:1810.06944 arXiv.org/abs/1810.06944
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Mio Murao ([村尾 美緒) now moves on to maps with indefinite causal orders for N∈{1,2,3} It helps for d=2, N∈{2,3},
not for N=2, d=3
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Mio Murao ([村尾 美緒) also has controllization algorithm, with U^(1/d)⊗d, also using antisymmetric states and control swaps. A large ancilla system is required.
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Mio Murao ([村尾 美緒)’s papers on this talk:
• arXiv:1810.06944 arxiv.org/abs/1810.06944
• arXiv:1808.05788 arxiv.org/abs/1808.05788
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